EQUILIBRIUM AGENDA FORMATION
Bhaskar Dutta,
Matthew O. Jackson
And
Michel Le Breton
No 628
Equilibrium Agenda Formation
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
¤September 2001
Draft: December 11, 2001
Abstract
We develop a de¯nition of equilibrium for agenda formation in general voting settings. The de¯nition is independent of any protocol. We show that the set of equilibrium outcomes for any Pareto e±cient voting rule is uniquely determined. We also show that for such voting rules, if preferences are strict then the set of equilibrium outcomes coincides with that of the outcomes generated by considering all full agendas for voting by successive elimination and show that the set of equilibrium outcomes corresponds with the Banks set. We also examine the implications in several other settings.
Keywords: Agenda, Equilibrium, Voting
JEL Classi¯cation Numbers: D71, D72
¤Dutta is at the Indian Statistical Institute, New Delhi 110016, India, and the
1
Introduction
The importance of agenda formation in a wide variety of settings, ranging from committees to popular elections, is self-evident. In fact, in some leg-islative settings where the voting on speci¯c bills is highly predictable, one might argue that the most interesting strategic interaction takes place in the formation of the agenda.
Indeed, the wide literature that analyzes various aspects of voting recog-nizes the importance of the agenda, and has shown how important it can be (e.g., McKelvey (1976, 1979)). Nevertheless, we still lack tractable models of agenda formation, and a detailed understanding of how the formation of the agenda ultimately a®ects the outcome of voting. To quote Ordeshook (1993):
More problematic is the issue of endogenous agendas, the process whereby agendas are formed via the sequential introduction and labeling of alter-natives to be voted on. ... The particular problem is that to apply game theory we must provide a game form that speci¯es precisely the identity of decision makers, the sequence with which they make decisions, and the information at their disposal when they act. And although agenda voting, like simple descriptions of elections, lends itself readily to the construction of such form, the processes whereby agendas are formed is far less structured and, thereby, less amenable to unambiguous game-theoretical analysis.
Ordeshook's statement points out the di±culty of modeling agenda formation stemming from the lack of a clearly de¯ned game form.
In this paper we provide a model of agenda formation, and in particular one that does not rest on a speci¯c game form or protocol. The way in which we do this is to examine the continuation equilibria that might extend from any given agenda. We do this inductively, de¯ning sets of possible contin-uations from any given agenda up to some point, and imposing sequential rationality.
equilibrium: It is an equilibrium to stop at some agenda only if no agent prefers any continuation equilibrium. We show that the sequential rational-ity and stopping conditions alone provide strong conclusions for what the set of equilibrium agendas can include.
In particular, we show that if a voting rule always selects an outcome that is Pareto e±cient relative to the agenda, then sequential rationality and stopping conditions imply that equilibrium agendas will result in voting out-comes that are Pareto e±cient overall. Moreover, one of our main results states that for Pareto e±cient voting rules the equilibrium outcomes will al-ways be a subset of what might arise from considering the set of complete agendas (including all outcomes). This result turns out to allow us to make fairly sharp predictions concerning equilibrium agendas in many settings. For example, if the voting rule does not depend on the speci¯c order of the agenda, then equilibrium agendas result in a unique outcome which is that when all alternatives are included in the voting. This also has important im-plications for voting rules where the order of the agenda does matter, such as the well-studied example of voting by successive elimination. There we show that equilibrium agendas always result in outcomes that lie in the Banks' set. Similarly, for voting rules that always pick outcomes that lie in the top cycle of the alternatives on the agenda, we show that the equilibrium agendas must result in outcomes that lie in the top-cycle of all alternatives. So, if for instance, a Condorcet winner exists and the voting rule is Condorcet consis-tent, then all equilibrium agendas include (and thus result in) the Condorcet winner.
Some Related Literature
Part of the motivation behind our analysis comes from the literature on \chaos" theorems. For instance, McKelvey (1976, 1979) has shown that in the context of majority rule and Euclidean settings, the top cycle of alternatives is either a singleton (a Condorcet winner) or the whole space. And, as Plott (1973) has shown, the second case is the generic one.1 This implies that in most cases, starting from one alternative one can ¯nd a sequence of alternatives leading to any other, where each one in the sequence beats the previous one. While the conclusion that one should draw from such a result and whether or not \chaos" is an appropriate nickname has been debated, it is clear that such a result makes it critical to have an understanding of equilibrium agendas; as otherwise one is left without any prediction. This is essentially the primary motivation for our analysis. As such, we come back below to examine the predictions our equilibrium notion makes in the context of voting by successive elimination, and discuss the relation to chaos theorems.
An alternative approach to modeling agenda formation is to assume a speci¯c protocol and analyze its implications. For instance random recogni-tion rules were studied in the context of multilateral bargaining (divide-the-dollar games) by Baron and Ferejohn (1989) (and the literature that fol-lowed). That approach provides for strong analytical conclusions. However, that approach is not so tractable outside of the distributive setting in which it is posed. Moreover, there are many applications where the protocol is not clear, as the above quote of Ordeshook points out. The advantages to the approach taken in this paper are that it can be applied to a general class of voting problems, where for instance, Euclidean preferences may not be appropriate; and it makes protocol-free predictions.
With regards to making protocol-free predictions, we remark that the sets of equilibria uncovered here should be viewed as a set of potential equilibria. Adding more knowledge of the speci¯c protocol may induce selections from the set we identify, and result in more speci¯c predictions. Nevertheless, as we shall show, fairly minimal requirements on the equilibrium set already allow for some tight predictions in the context of a variety of voting rules. Thus there are important aspects of equilibrium agendas that can be characterized
1See Austen-Smith and Banks (1999) for a nice discussion of this literature and
without detailed knowledge of the protocol.
Work on equilibrium agenda formation has also been done in other con-texts. For example, Banks and Gasmi (1987) examined equilibrium agenda formation in three person committees. Their analysis is of a Euclidean set-ting and one where the three committee members can make only one proposal each, and so agendas are truncated. Specifying the problem to this level leads to sharp predictions. More recently, Penn (2001), in the context of three per-son divide-the-dollar games has extended the analysis to allow for arbitrary agenda lengths by a clever adaptation to in¯nite agendas, and shows that sharp predictions again result (but di®er from those of Banks and Gasmi). The above results are very encouraging in the face of \chaos" theorems, and may be thought of as answering those theorems by saying that if we do model agenda formation, then we can make speci¯c predictions. Nevertheless, the above analyses come in very speci¯c settings and are dependent upon the geometry of Euclidean preferences, and in some cases having three proposers and having a strong symmetry among them. Our analysis attempts to pro-vide an equilibrium de¯nition that can be applied to a more general set of problems. Our main motivation is to develop a concept that does not require such speci¯c geometry, and at the same time does not demand detailed spec-i¯cation of the proposal protocol. 2 As such, the predictions our analysis makes are not always as crisp; but nevertheless are fairly speci¯c in many settings.
Equilibrium agenda formation has also been analyzed in another setting. That is the setting of strategic-candidacy. For instance, in Osborne and Slivinski (1996) and Besley and Coate's (1997) models of citizen- candidates, the decision to enter an election and take a position is studied under equilib-rium. In other work (Dutta, Jackson, and Le Breton (1998, 2001)) we have examined the properties of equilibrium sets of candidates for a variety of vot-ing rules and for votvot-ing by successive elimination. While the issue of strategic candidacy is an important example of endogenous agenda formation, mod-eling agenda formation more generally requires a di®erent approach. Most
2Another distinction is that our approach is based on one of inductively de¯ning
importantly, the candidacy decision ultimately rests with the candidate.3 This means that the proposal abilities of agents are limited. This provides for di®erent strategic considerations than, for instance, in a legislative setting where proposers are not restricted in the alternatives that they may propose. Below, we compare the outcomes of strategic agenda formation in the con-text of strategic candidacy and in the more general setting where proposers are not limited; and see that there are important distinctions.
Another branch of the literature that has touched on equilibrium agenda formation is that which has looked at sophisticated voting by successive elim-ination. In particular, a de¯nition of equilibrium agendas appears in work by Miller, Grofman, and Feld (1990). In their analysis an agenda is an equilib-rium if nobody would gain by adding some alternative to the current agenda. The important di®erences between such a de¯nition and the ones presented here are in the beliefs of the proposers. The de¯nition just described does not account for the fact that in many cases the agenda will not end, but instead will be subject to further modi¯cations. Thus, proposers are act-ing myopically.4 If proposers can make any predictions about continuations, rather than myopically assuming the agenda will end, then the outcome could be quite di®erent. This emphasizes an important aspect of our de¯nitions. Incorporating such sequential rationality and anticipating equilibrium con-tinuations is the foundation on which we build our de¯nitions. We come back to examine the impact of this feature below, when we apply our de¯nitions to voting by successive elimination.
Finally, we mention a distantly related literature in terms of applications and speci¯cs; but more closely related in terms of ¯nding equilibrium de¯-nitions that are not tied down to protocol speci¯cation. In particular, the literature on coalition formation (and on coalitional bargaining) faces a sim-ilar di±culty to that expressed in the quote of Ordeshook above. Writing down speci¯c bargaining protocols allows for sharp predictions, but ones that
3Even if one allows candidates to be nominated, they usually have the option to decline
to run.
4Austen-Smith (1987) and Groseclose and Krehbiel (1993) also examine equilibrium
may not be robust and are not so easily adapted to settings where the pro-tocol is not obvious. Chwe (1994) provides a de¯nition of consistent sets of alternatives that might come out of coalitional bargaining settings, that is not dependent on any speci¯c protocol and yet still makes intuitively appeal-ing predictions in many applications. Our approach here is intended to do the same thing for agenda formation problems. While there is a parallel in spirit, the actual equilibrium de¯nition that we provide and the issues we face bear little resemblance to that in Chwe's work.5
2
De¯nitions
Alternatives
There is a set of alternatives X. Generic elements are denoted x, y, and
z.
We begin the analysis with the case whereX is ¯nite and with #X =m, as this brings out the intuitions most clearly. We then return to show how our analysis extends to the in¯nite case in Section 5.7.
Society will select one of these alternatives. These may be potential bills that a legislature might enact, a set of candidates that a society might elect, or a list of potential decisions that a committee might reach.
Voters or Decision Makers
The set N =f1; : : : ; ngis a ¯nite set of voters.
These are the individuals who are involved in determining the agenda and the outcome from that agenda. In Section 6 we discuss the possibility of having special roles for some individuals.
Preferences
Individuals have preferences over the set of alternatives represented by a complete and transitive binary relation, Ri. The strict preference relation associated with Ri is denoted Pi, and is de¯ned by xPiy if and only if not
yRix. As usual, knowingPi similarly de¯nesRi, and so we keep track of the strict relationship with the weak one being inferred.
5As a note, our use of the word consistency has no relationship to that of Chwe's
Let P denote the set of admissible pro¯les of preference relations. The notationP 2 P denotes a generic pro¯leP = (P1; : : : ; Pn).
In some applicationsP will be a restricted domain. A number of di®erent examples appear in what follows.
Agendas
An agenda of length k 2 f1; : : : ; mg is a ¯nite vector of alternatives (x1; : : : ; xk)2Xk, with the restriction that xi 6=xj for each i6=j.
LetAk denote the set of agendas of lengthk, and let A=[m
k=1Ak be the set of all agendas.
The restriction that the same alternative not appear more than once in an agenda is common to many legislative and committee settings. Given that the set of alternatives X could be quite large and dense, this does not prevent an alternative and a close approximation of it from appearing in an agenda.
Depending on how the voting procedure works, the sequence of the agenda may or may not matter. For instance if the agenda is simply a list of nomi-nated candidates and some neutral voting procedure is used, then the agendas (x; y; z) and (z; y; x) would be equivalent. However, if the voting procedure is non-neutral, then the sequence can be important. For instance, under voting by successive elimination where proposed alternatives are voted upon in reverse order of their proposal the agendas (x; y; z) and (z; y; x) are not equivalent and could lead to di®erent outcomes.
Extensions of an Agenda
In many situations of interest, some part of an agenda will already be on the table. For example, if there is a status quo, then it may take the ¯rst place in any agenda that follows. More generally, in building a de¯nition of equilibrium we need to be able to make predictions starting from various existing agendas and so it is useful to consider the concept of the extensions of a given agenda.
With this in mind, for any k and a 2 Ak let A(a) to be the set of all agendas that agree with ain the ¯rst k spots. That is,
A(a) =fa02A j a0h=ah 8h2 f1; : : : ; kgg:
A voting procedure is a function V :A£ P ! X such that V(a; P)2a
for all a 2A and P 2 P.
A voting procedure thus summarizes the choice the society would make from a given agenda at a given preference pro¯le. This formulation is very °exible and allows for many applications. For instance, it could be that
V is determined by strategic voting or instead by sincere voting. Also, V
might depend on the ordering of the agenda or it might not; andV might be anonymous, or it might treat some voters specially.
The details of how V is determined will not be important in developing our de¯nition of equilibrium agenda formation. Later, in providing some results about the properties of equilibria, we will specify some properties of potential voting rulesV and examine some speci¯c voting rules.
3
Equilibrium Agendas
Before presenting the formal de¯nitions of equilibrium, we begin with a sim-ple examsim-ple to motivate and illustrate the de¯nitions.
Example 1
X =fx; y; zgand x is the status quo. The voters' preferences form a classic cycle:
² xP1yP1z
² yP2zP2x
² zP3xP3y
Herex beatsy, y beatsz, and z beats x under majority rule.
The voting rule is sincere voting by successive elimination. For instance, if the agenda is (x; z; y), then ¯rst a vote is held between y and z, and then the winner is matched against x. Under sincere voting, the outcome of this agenda would be x, as y would defeat z and then x would beat y.6 Here,
6A situation which approximately ¯ts this one is that of the Powell amendment
the only possible outcomes are x from agendas (x; y; z), (x; z; y), (x; y) and
x; and z from agenda (x; z).
Let us discuss equilibrium conditions based on this example. Once an agenda of three alternatives has been reached, there are no alternatives left to propose, and so an equilibrium continuation is simply the agenda in question. Next let us step back and consider an agenda of length 2 that starts with the status quox. There are only two such agendas to consider. One is the agenda (x; z). If this agenda is reached, then agent 1 by adding the alternative y
would change the outcome from z to x. This would make agent 1 better o®, and so the agenda (x; z) would not be stable to amendment.7 This suggests one of the conditions in our equilibrium de¯nition: that stopping at a given agenda is an equilibrium if and only if there is no agent who can bene¯t from advancing the agenda to some further continuation equilibrium. So, the only continuation equilibrium following (x; z) is the agenda (x; z; y). Next, let us back things up. Given the agenda x in place, if some agent proposes z next, then if she should anticipate that the result will be the full agenda (x; z; y) with outcome x. This embodies another part of the equilibrium de¯nition: agents should anticipate equilibrium continuations from extensions of an agenda. In this case, no matter what happens afterx, any continuation equilibrium must lead to the outcome ofx. This actually means that stopping at x can be an equilibrium. Whether or not the other agendas that lead to x are also included as equilibrium continuations from
x, is something that is not mandated by our basic de¯nitions of equilibrium.
in the House of Representatives was one that would introduce some federal funding of local public schools. The amendment to the billy introduced by Powell was to deny federal funding to public schools that practiced segregation (this was in the 1950's). As Denzau, Riker and Shepsle argue, sincere voting could be explained by the di±culty in explaining voting against the Powell amendment to one's constituency. In fact, the situation had some mixture of sincere and sophisticated voting, as some representatives who opposed funding (and supported segregation) may have voted for the Powell amendment in the ¯rst round and then against it in the second round. So there may have been some conservative representatives who had the preferences of voter 1 except withz andyreversed, but who when voting strategically would vote the same as voter 1 would vote when voting sincerely.
7Interestingly, in this example if we require a second agent to support a proposal in
However, a further consistency condition that we add would imply that the other agendas leading tox would also be equilibria in this example.
With some of the basic ideas from this simple example in hand, let us now consider the full de¯nition of equilibrium agendas.
First, notice as in the above example, de¯ning behavior at one agenda requires having some notion of what will happen following various extensions of the given agenda. Thus, the de¯nition involves sets of continuation equi-libria to be de¯ned from each starting point. This is necessarily a set of sets, where a set of continuation equilibria is speci¯ed starting from each possible agenda.
We deliberately impose only weak requirements in de¯ning equilibrium sets. Although taking such an approach allows for various collections to satisfy the de¯nition, these weak requirements already have substantial im-plications for which outcomes might be reached.
A collection of sets of continuation equilibria for a pro¯le of preferences
P 2 P is a collection fCEV(a; P)ga2A, where C EV(a; P) ½ A(a) for each
a2A, that satis¯es the following properties. GivenfCEV(a; P)ga2A, let
CV+(a; P) =[x =2aC EV((a; x); P):
SoCV+(a; P) is the set of all continuation equilibria that could result if some alternative is added to an existing agendaa.8
A continuation equilibrium set satis¯es the following for eacha 2A: (CE1) (Equilibrium Continuations) CEV(a; P) is a nonempty subset of a [
CV+(a; P) and
(CE2) (Stopping Requirements)a 2CEV(a; P) if and only ifV(a; P)RiV(a0; P) for all a02CV+(a; P) and for all i2N.
Part (CE1) is a sequential rationality condition that simply says that the possibilities from any agendaa are either to stop at a, or to add a new al-ternative to the agenda and then follow some continuation equilibrium from
8We remark that ifa2Am, thenC+
V(a; P) =;. Under (CE1) and (CE2) below, this
the resulting agenda. This is a condition that essentially just requires that the sets of equilibria for di®erent agendas have some minimal relationship to each other: if agents anticipate thata0= (a; x; : : :) is a continuation equilib-rium starting at a, then they must also expect it to still be a continuation equilibrium when they have reached (a; x).
Part (CE2) describes conditions under which it can be an equilibrium for agents to `stop' ata. If every agent ¯nds that V(a; P) is at least as good as the outcome corresponding to any other possible continuation equilibrium, then no agent has an incentive to extenda. Conversely, if some agenti¯nds the voting outcome corresponding to some continuation equilibrium strictly preferred to V(a; P), then thisi will rather make a proposal and follow the preferred continuation equilibrium, and the agenda will not stop at a.
One of our main themes developed below is that these minimal conditions already have some very strong implications and imply a great deal about sets of equilibria.
While imposing some restrictions on collections of sets of continuation equilibria, conditions (CE1) and (CE2) can still allow for a multiplicity of collections of equilibrium continuations that satisfy the de¯nition. Essen-tially, (CE1) and (CE2) give us some weak limitations on what can be in the set of equilibria, but they do not tell us much about which agendas must be included in the set. Consistency (CE3), below, addresses this issue.
We say that an agendaa0= (a; x; : : :) 2CV+(a; P) isrationalizableif there exists i 2 N and a00 2 C EV(a; P) with either a00 = (a; y; : : :) with y 6= x or
a00=a such thatV(a0; P)RiV(a00; P).
The idea of rationalizability is thati proposes adding x to the agendaa
under the belief that it will result in the agenda a0, and that if i does not propose adding x then instead the continuation would be a00. As a00 is a continuation equilibrium, this belief can be justi¯ed.
We say that a collection of sets of continuation equilibria is consistent if it satis¯es
(CE3) (Consistency) If a0 2CV+(a; P) is rationalizable, then a0 2 CEV(a; P). Conversely, if a0= (a; x; : : :)2C E
V(a; P) and eithera 2CEV(a; P) or
are rationalizable, subject to two exceptions. One is that stopping is han-dled under (CE2), and so the rationalization ofa itself is already addressed. The second is that an equilibrium continuation agenda does not need to be rationalizable if it is a \unique" equilibrium continuation. Note that in this second case, the ¯rst part of the condition implies that all agents unani-mously ¯nd the outcomes under (a; x; : : :) preferred to stopping or adding any other alternative to a.
Later, we come back to discuss other notions of rationalizability and consistency.
We point out some important aspects of the above de¯nitions.
First, the de¯nitions necessarily involve a whole collection offCEV(a; P)g, one set for each a2 A. This re°ects the forward-looking aspect of the de¯-nition. In order to know what is an equilibrium starting at one agenda, one has to be able to anticipate what will happen starting at extensions of that agenda.9
Second, there always exists at least one collection fCEV(a; P)g satisfy-ing (CE1)-(CE3), which is easily seen via a backwards induction argument, starting with agendas of full length, and then working back to smaller agen-das.
Third, the set CEV(a; P) is not always uniquely determined. That is, there may be several di®erent sets which satisfy conditions (CE1) and (CE2); even when consistency (CE3) is imposed. This stems from the fact that the conditions are designed to be weak, to specify conditions that an equilibrium set should satisfy, but not so strong as to always uniquely determine that set. Again, this traces back to our deliberate avoidance of any reliance on an ad hoc formulation of the proposal process. To see an easy example of the potential multiplicity of equilibrium continuations, consider a somewhat degenerate voting rule as follows.
Example 2 Multiple Collections of Sets of Continuation Equilibria:
Under V the outcome is always the second alternative proposed in the agenda (or the ¯rst if the agenda is a singleton), regardless of the preference pro¯le. SoV(a; P) =a2 ifa 2Ak withk ¸2 and V(a; P) =a1if a2A1.
9Of course, this is similar to a de¯nition such as subgame perfect equilibrium where
This is a peculiar voting rule, but one that allows for a simple illustration of the multiplicity of equilibria. Note that in this case, C EV(a; P) = A(a) is uniquely determined for any a 2 Ak for k ¸ 2. This follows since once the second alternative has been proposed the outcome is already determined and the rest of the agenda is completely irrelevant and so under (CE2) and (CE3) all continuations are then equilibria. Now consider the outcome that is proposed in the second place in the agenda. In particular, letX=fw; x; y; zg
and consider a preference pro¯le where some agents have preferences z, y,
x, w, and others have preferences z, x, y, w; where the ordering speci¯es the strict preferences wherew is the worst alternative. Consider starting at the agenda a=fwg. So,w is the status quo. Conditions (CE1) and (CE2) have only very weak implications here: it cannot be an equilibrium to stop at fwg. Beyond that, they allow for a variety of continuation equilibrium sets. Once consistency is added, however, things are tied down to a greater degree. In particular, there are two sets which satisfy (CE1), (CE2) and (CE3). The ¯rst such set consists of all extensions of a with z in second place (i.e., CEV(a; P) = A((w; z))); and the second such set consists of all extensions of a with any of x, y, or z in second place (i.e., CEV(a; P) =
A((w; x))[A((w; y))[A((w; z))).
In this example, consistency (CE3) still does not uniquely tie things down. One might argue that extensions of (w; z) are really the only sensible equilib-rium continuations in the above example, as they are unanimously preferred to proposalsx andy. One may wish to impose such additional conditions on the notion of equilibrium (and we discuss this more fully in Section 6). How-ever, as we shall see, if we restrict attention to more sensible voting rules, such as those which satisfy a Pareto e±ciency condition, consistency will already tie things down uniquely without the imposition of any additional conditions.
4
Equilibrium Agendas for Pareto E±cient
Voting Rules
An alternative x 2 B ½ X is Pareto e±cient relative to P and B if there does not existy 2B such thatyRix for all i2N andyPjx for some j 2N.
V is Pareto e±cient if V(a; P) is Pareto e±cient relative to P and the alternatives ina, for each a2A andP 2 P.
Given a collectionfCEV(a; P)ga2Aand anya 2A, let P EV(a; P) denote the set of agendas in CV+(a; P)[athat result in Pareto e±cient alternatives (considering all of X).
Theorem 1 For any Pareto e±cient voting rule V and preference pro¯le
P 2 P and collection of sets of continuation equilibria fCEV(a; P)ga2A,
V(a0; P) is Pareto e±cient (considering all alternatives) for all a and a0 2 C EV(a; P). 10 Moreover, if consistency is satis¯ed, thenfCEV(a; P)ga2A is uniquely de¯ned and described by
CEV(a; P) = (
P EV(a; P) if V(a; P)RiV(a0; P) 8i and a02CV+(a; P)
P EV(a; P)na otherwise.
The ¯rst result in Theorem 1 is that equilibrium agendas of Pareto e±-cient voting rules must result in outcomes that are Pareto e±e±-cient overall. This conclusion is not quite as obvious as it seems. For instance, it could be that x is Pareto dominated by y, but that V(a0; P) 6= y for all a0 2 A(a). This means that since y is never in the range of V, it does not threaten
x. The proof uses the fact that if y is added to an agenda containing x, then the outcome cannot bexand must instead be some other outcome that some voter prefers to x. Building on this reasoning we rule out equilibrium agendas leading tox. The details are provided in the proof in the appendix. The second result in Theorem 1 is that under consistency the continua-tion equilibria of Pareto e±cient voting rules are uniquely determined and described by a simple algorithm.
10Theorem 1 also holds if one replaces Pareto e±ciency everywhere by weak Pareto
e±ciency, where an alternativex2B½X is weakly Pareto e±cient relative toP andB if there does not existy2B such thatyPixfor alli2N. This weakens the assumptions
The implications of Theorem 1 are even stronger when preferences satisfy a mild restriction.
LetP¤ be the set of all pro¯les satisfying the restriction:
8x; y 2X;9i 2N such thatxPiy oryPix:
So, P¤ is the set of pro¯les such that it is never the case that all indi-viduals are indi®erent between some pair of alternativesx; y. Of course, this condition is satis¯ed when individual preferences are strict, but also holds more generally including where some transfers or distribution of resources are possible. In this case, we obtain a characterization of continuation equi-librium outcomes that does not even require an inductive de¯nition.11
Theorem 2 Consider a Pareto e±cient voting rule V and pro¯le of pref-erences P 2 P¤. If fCEV(a; P)ga2A is a collection of sets of continuation equilibria, then the outcomes corresponding to continuation equilibria follow-ing some agendaaare a subset of those that can be found by considering only full length agendas that are extensions of a. That is,
[a02CEV(a;P)V(a0; P)½ [a02A(a)\AmV(a0; P):
If in addition consistency is satis¯ed, then these sets are equal:
[a02CEV(a;P)V(a0; P) =[a02A(a)\AmV(a0; P):
Theorem 2 shows how powerful the implications of the simple stopping condition are. It states that the equilibrium outcomes correspond to those where complete agendas are considered. The idea behind this follows an inductive proof. Suppose this is true once an agenda is of lengthk or more. Now suppose that some agenda of lengthk¡1 is an equilibrium agenda and results in an outcome that di®ers from all full length agendas, and thus all continuation equilibria if any outcome is added. Given Pareto e±ciency, some agent must prefer some outcome of a longer agenda that is a continuation
11To see an example of why this condition is needed in the theorem, consider a situation
equilibrium if some alternative is added to the current agenda to that of stopping. Then (CE2) implies that stopping cannot be an equilibrium.
The proof of Theorem 2 is in the Appendix. The second half of the proof actually follows from a stronger claim which does not invoke Pareto e±ciency of the voting procedure. Since this is of independent interest, we state it here.12
Claim 1 For any voting procedure V, preference pro¯le P 2 P and a2 A, iffC EV(a; P)ga2A is a collection of sets of continuation equilibria satisfying consistency, then any Pareto e±cient alternative that can be reached via some full length continuation ofais an equilibrium continuation outcome following
a at P.13
5
Applications to Speci¯c Voting Rules and
Settings
In order to demonstrate the implications and usefulness of Theorems 1 and 2, we apply them to a number of settings including some prominent ones.
5.1
Order Independent Voting Rules
A voting ruleV is order independent if V(a; P) = V(a0; P) whenever fx 2 ag=fx2a0g.
Order independent voting rules are those for which the ordering of the agenda does not matter. Neutral voting rules are order independent, but there are also important order independent voting rules that are non-neutral. Consider the following example: candidates are people who are seeded ac-cording to their age (or experience, rank, etc.). Regardless of the order in which they are proposed or nominated, the two youngest candidates are voted
12In fact we prove stronger statements in the appendix, showing that even for ine±cient
voting rules there is a minimal consistent set of equilibria (in terms of set inclusion), which corresponds to the de¯nition under the algorithm above. It is under Pareto e±ciency that this must coincide with all consistent sets of equilibria.
13Since we show in the appendix that there is a minimal consistent set of equilibria (in
upon, then the winner of that vote is pitted against the next youngest, etc.. This rule is independent of the order in which the candidates are proposed, and yet it is still a sequential rule and is clearly not neutral. Therefore, we emphasize that \order independence" refers only to the order of the agenda and does not mean that the voting rule itself is not based on some implicit ordering of alternatives.
Note that for any order independent voting rule, V(a; P) =V(a0; P) for any a and a0 in Am. With an abuse of notation, we write this outcome as
V(X; P).
The following is a direct corollary of Theorem 2.
Corollary 1 For any Pareto e±cient and order independent voting rule
V, preference pro¯le of preferences P 2 P¤, collection of sets of continuation equilibria fC EV(a; P)ga2A (i.e., satisfying (CE1) and (CE2)), and agenda
a2A, there is a unique continuation equilibrium outcome
[a02CEV(a;P)V(a0; P) =V(X; P):
An important remark about Corollary 1 is that it does not require con-sistency, but follows from (CE1) and (CE2) in the de¯nition.
The following example shows how Borda's rule is covered under Corollary 1.
Example 3
Voters' preferences are :
² xP1wP1yP1z
² xP2wP2yP2z
² zP3wP3yP3x
This is a Pareto e±cient and order independent voting rule.
It is easily checked that w wins whenever it is on the agenda. Also, x
wins if it is present butw is not. If just y and z are present, then y wins. z
only wins if it is the only proposed alternative.
Corollary 1 implies that the outcome of any equilibrium agenda must be
w in this example. Indeed, it is easily seen that no agenda leading toy or z
can be an equilibrium, as addingwto the agenda will lead to a continuation equilibrium outcome ofwwhich would be preferred overyorzby some agent. Similarly, if an agenda leads to x, then adding w will lead to a continuation equilibrium ofw, which is better for voter 3 thanx.
5.2
Tournaments and Top Cycle Consistent Rules
The following de¯nitions are useful in some of the remaining applications.
Tournaments
In many contexts, the preferences of the voters can be summarized (even for strategic purposes) by the majority voting relation that is induced over pairs of alternatives. A tournament is a binary relation that summarizes the important aspects of voters' preferences in some contexts.14 More formally, the majority voting tournament is de¯ned as follows.
GivenP 2 P, denote by T(P) the binary relation de¯ned by
xT(P)y , #fi2N :xPiyg>#fi2N :yPixg
T(P) is always asymmetric and if nis odd and individual preferences are strict thenT(P) is complete. If we break ties in some deterministic manner, then even in cases with an even number of voters or indi®erencesT(P) is also complete, and therefore a tournament (an asymmetric and complete binary relation). In what follows, unless speci¯ed otherwise, we will assume that ties are broken so thatT(P) is complete. T(P) is referred to as the majority tournamentinduced by P.
The Top Cycle
14See Laslier [11] for an illuminating account of the principal results in the vast literature
As the majority tournament is not necessarily transitive, it can have cycles. A prominent cycle that we refer to in the sequel is the top cycle associated with a tournament.
The top cycle of T(P), denoted by T C(X; P) is the set fx2 X : 8y 2 X;9x1; : : : ; xk in X such thatx1=x; xk= yand xiT(P)xi+18i= 1; : : : ; k¡ 1g i.e. the set of alternatives that can reach any other alternative in X via a T(P)-chain of arbitrary length. For subsets of alternatives, B ½ X, there is a corresponding de¯nition and we denote that set T C(B; P). When there is noB ½ X indicated, then we are referring to the top cycle relative to X, and we use the notation T C(a; P) to denote the top cycle relative to the set of alternatives in the agenda a under the tournament T(P).
A voting rule istop cycle consistentat aP such thatT(P) is a tournament ifV(a; P)2T C(a; P) for any a2A.
Condorcet Winners and Consistency
An alternativefxgis a Condorcet winner relative toB ½XifT C(B; P) =
fxg. That is, a Condorcet winner is an alternative that beats every other alternative in B underT(P).
A voting rule V is Condorcet consistent if V(a; P) selects a Condorcet winner whenever one exists relative to T(P) and the alternatives in a.
5.3
Equilibrium Agendas for Top Cycle and Condorcet
Consistent Voting Rules
If the voting procedureV arises from strategic voting on a binary tree, then it follows from McKelvey and Niemi (1978) that V is top cycle consistent. Thus, the following proposition covers a wide variety of applications.
Proposition 1 Consider aP such thatT(P)is a well-de¯ned tournament and a collection of sets of continuation equilibriafCEV(a; P)ga2A(i.e., sat-isfying (CE1) and (CE2)). IfV is top cycle consistent, then all equilibrium outcomes following any agenda are in the (overall) top cycle. Moreover, if V
is Condorcet consistent and there exists a Condorcet winnerx atP, then all of the equilibrium continuations from any agenda lead to x.
1 is straightforward for the case where the preference pro¯les inP¤are strict and the voting rule is Pareto e±cient. Then, from Theorem 2 we know that the equilibrium outcomes coincide with those that are full agendas and extensions of the starting agenda. These must select from the top cycle. The proof when the preference pro¯les are not necessarily in P¤ or the voting rule is ine±cient is slightly more complicated, as then Theorem 2 cannot be applied. The proof is still relatively short and appears in the appendix.
A direct corollary of Proposition 1 is that all equilibrium agendas in a setting with single-peaked preferences and a Condorcet consistent voting rule lead to the outcome of the median of the voters' peaks.
5.4
Voting by Successive Elimination and Equilibrium
Agendas
The voting procedure ofvoting by successive eliminationis de¯ned as follows. Consider some agenda a 2 A and let a = (x1; : : : ; xk). In the successive elimination procedure, a vote is ¯rst taken to eliminate either xk or xk¡1. The `winning' alternative from the ¯rst round is compared to xk¡2, and a vote is taken to eliminate either surviving alternative from the ¯rst vote or
xk¡2, and so on. After (k¡1) comparisons, the last surviving alternative is declared to be the voting outcome.
At each stage, the elimination of one alternative is according to majority voting. This is well-speci¯ed when T(P) is complete. However, in cases where there are ties under the majority preference relation, either resulting from personal indi®erences or from an even number of voters, T(P) is not complete. In this case, voting by successive elimination needs to be more completely speci¯ed.
We do so as follows. At each stage allow individuals to vote for one of the two alternatives or to abstain (in the case where they may be indi®erent). In case of a tie in the voting between alternativesxi and xj, xi is elected if and only if xi comes before xj in the ordering of voting (i < j). This favors alternatives proposed earlier in the agenda under ties, which is a natural way to break ties (given that they have not already been broken underT(P)).
At the last stage of voting, if the voting boils down to a comparison of
x and y where xprecedes y in the successive elimination procedure, then x
However, in order to determine the eventual voting outcome, it is also necessary to describe how voters act. We ¯rst examine the case where they vote strategically at each stage, and so focus on the sophisticated voting out-come of this binary voting procedure. This is the outout-come under the iterative elimination of weakly dominated strategies that has been well-studied (see Shepsle and Weingast (1984) for the algorithm identifying the outcome). 15
Let S(a; P) denote the sophisticated voting outcome under voting by successive elimination on agendaa.
The Banks Set
The Banks setassociated with the tournament T(P), denoted BS(P), is de¯ned by
BS(P) =[a2AmS(a; P):
Thus, the Banks set is the set of sophisticated voting outcomes under voting by successive elimination under all possible full agendas.
There are situations, however, where some orderings ofXare not relevant. For instance,X may contain a distinguished alternative x1which acts as the status quo. In many legislative procedures, the status quo is treated as if it were the ¯rst proposal in the agenda. Recall that our tie-breaking rule in caseT(P) is not complete naturally privileges the status quo against the amendments. We generalize the de¯nition of the Banks set in the following way.
Given any a2Ak, let
BS(a; P) = [a02A(a)\AmS(a0; P):
Equilibrium Agendas and Voting by Successive Elimination
Given that voting by successive elimination is a Pareto e±cient voting rule, we have the following corollary of Theorem 2.
15The Shepsle-Weingast algorithm was de¯ned for the case whereT(P) is complete. Our
Corollary 2 Consider a collection of sets of continuation equilibriafCEV(a; P)ga2A (satisfying (CE1) and (CE2)) and any pro¯le of preferences P 2 P¤. For all
a2A,
[a02CEV(a;P)S(a0; P)½ BS(a; P);
and if consistency (CE3) is also satis¯ed, then
[a02CEV(a;P)S(a0; P) = BS(a; P):
Note that the result above also holds if we set the starting agenda a to be the emptyset.16
Corollary 2 states that not only does the Banks' set capture the set of outcomes that could arise from arbitrary full length agendas, but that these are also precisely the set of potential equilibrium outcomes when the agendas are endogenous.
While Corollary 2 provides a precise characterization of equilibrium agenda outcomes for an important voting procedure, it is still useful to show that this characterization completely ties down the outcome in some interesting cases. We now show this in the context of an interesting \pork barrel poli-tics" setting. In particular, even though in some cases the top cycle of the majority voting relation may be very large, the Banks set, and thus the set of equilibrium agenda outcomes, can be a singleton.
5.5
Voting over Projects
Ferejohn, Fiorina, and McKelvey (1987) consider the following model. N
is a set of legislators (with n odd), each of whom has a project for their constituency. The projects have value only for their constituents, but the cost of a project, if it is undertaken, is split evenly among all constituencies.17 Ferejohn, Fiorina, and McKelvey assume that projects have di®erent costs, so as to ensure thatT(P) is complete, but that is not assumed here (as we can extend their result given our procedure for breaking ties).
16An easy way to see this is simply to extend the set of alternatives to include somex 0
such that all alternatives are preferred tox0by all agents underP, and then seta=fx0g
and then apply the theorem as it stands.
17This assumption is not necessary. All that matters is that the legislators agree about
So, this is a model of pure \pork-barrel" politics. Here the set of alterna-tivesX is simply a list of which projects are undertaken, and soX =f0;1gn. Voting over an agenda is done by sophisticated voting by successive elimina-tion.
Given this setting, legislators' preferences take a speci¯c form. Their favorite alternative is to have their own project undertaken and no other projects undertaken. Beyond the decision concerning a legislator's own project, he or she simply prefers to minimize the costs of the other projects under-taken. The critical freedom in the preferences is in the relative costs of projects, which determines which projects a legislator might tolerate being undertaken in conjunction with his or her own, before the cost becomes so high that he or she would prefer to have none built at all.
An interesting aspect of the Ferejohn, Fiorina, and McKelvey (1987) model is the importance of a status quo. The status quo is that no projects are undertaken. Applying our equilibrium approach to this model is of par-ticular interest as it shows how the status quo can tie down equilibrium agendas, and illustrates why we have been careful to de¯ned continuation equilibrium concepts that allow for a status quo. It also shows that the conclusions reached by Ferejohn, Fiorina, and McKelvey (1987) without an equilibrium analysis, are robust to an equilibrium formulation.
Let X¤(P) denote the set of x 2 X that (i) undertake exactly n+12 projects, (ii) are as cheap as any other choice of exactly n+1
2 projects, and (iii) are such thatxT(P)0.
Corollary 3 Consider any pro¯le of admissible preferences P 2 P and collection of sets of continuation equilibriafCEV(a; P)ga2A(satisfying (CE1) and (CE2)) in the extension of the Ferejohn, Fiorina, and McKelvey setting where some projects may have identical costs.
[a02CEV(a0;P)S(a0; P) = ½
X¤(P) if X¤(P)6=;
0 otherwise.
5.6
Sincere Voting and an Absence of Chaos
The previous results show that equilibrium conditions on agendas can make narrow predictions. The results concerning the Banks set and voting by successive elimination were constrained to sophisticated voting. As much of the literature on chaos theorems (e.g., McKelvey (1979)) was restricted to sincere voting we show that the same is true there. In particular, we show that even in situations where the top cycle is large (even the whole set of alternatives), considering only equilibrium agendas still narrows the set of predictions in well-de¯ned ways.
While the setting we consider in this section is a ¯nite one (see the next section for the in¯nite case), we can still see the essence of chaos theorems in the following way. Considersincere voting by successive elimination, where when asked to compare any two alternatives, voters vote for the one that they prefer, not anticipating the outcome of the votes yet to come in the sequence.18 19
The critical observation is that for any x2 T C(T(P)) and any k, there exists an agendaa2Ak, such thatV(a; P) =x, whereV is sincere voting by successive elimination. In particular, setting k =m, any x in the top cycle can be reached by at least one full length agenda (in fact at least two).20
18One might also term this myopic voting. Note, however, that this corresponds to
sophisticated voting under the following alternative voting rule. That is important, as otherwise there would a schizophrenia between sophisticated (forward looking) agenda formation and myopic voting, and this exercise would only serve as a comment on the chaos literature. The closely related voting procedure for which this is sophisticated is as follows. On an agenda a= (x1; : : : ; xK), selectx1 unless a majority votes to move on to
x2; then select x2 unless a majority votes to move on to x3, and so forth. Sophisticated
voting on this rule can be solved as follows. If one gets to the last decision of whether or not to select xK¡1 or move on, then the vote will be a sincere vote between xK and
xK¡1. Anticipating this, the previous vote is a sincere vote betweenxK¡2and the sincere
winner betweenxK andxK¡1. Rolling this back up the voting tree, this is solved exactly
as a sincere vote by successive elimination.
19Note that sophisticated behavior in voting by successive elimination can preclude some
alternatives from the top cycle as ever being equilibrium outcomes as we already saw in Corollary 2.
20A recipe is as follows. Find an ordering of the K alternatives in the top cycle x =
x1; x2; : : : ; xK, such that xiT(P)xi+1 for each i < K. Such an ordering always exists.
This means that if we are not able to do any selection over agendas, then any alternative in the top cycle can be an outcome.
The following example, however, illustrates that our de¯nition of equilib-rium selects from the agendas. Here only a subset of the top cycle alternatives are equilibrium outcomes, even though all alternatives (other than a unani-mously bad status quo) are in the top cycle. Thus, the notion of equilibrium does preclude alternatives and make selections from the top cycle.21
Example 4
Voters' preferences are :
² x5P1x2P1x3P1x4P1x1P1x0
² x4P2x5P2x1P2x2P2x3P2x0
² x3P3x4P3x5P3x1P3x2P3x0
The induced tournament T(P) is that
² x5 beatsx0, x1, x2, and x3,
² x4 beatsx0, x1, x2, and x5,
² x3 beatsx0, x1, and x4,
² x2 beatsx0 and x3,
² x1 beatsx0 and x2.
Note that hereBS(fx0g; P) =fx3; x4; x5gandT C(X; P) =fx1; x2; x3; x4; x5g; and also that bothx4 and x5 Pareto dominate x1.
Under sincere voting by successive elimination, the agendas (with a status quo ofx0) that can lead to an outcome ofx1are those that follow the ordering
the outcome.
21In light of Proposition 1, equilibria under sincere voting by successive elimination will
of the index of the alternatives without gaps, starting atx0, except possibly that the last two alternatives may be switched.22
None of these are equilibrium agendas when the status quo is a =fx0g (i.e., none of these are in CEV(fx0g; P)). Thus, x1 is not an equilibrium agenda outcome whenV issincere voting by successive elimination.
First, it is easily checked thatfx0; x1g andfx0; x2; x1g, are not continua-tion equilibrium agendas (i.e., stopping once they are reached), as addingx5 will lead to an outcome of either x4 or x5 which are unanimously preferred tox1; and so (CE2) is violated. Thus they could not be equilibrium agendas beginning at x0. We can also check that the agenda fx0; x1; x3; x2g is not a continuation equilibrium. If eitherx5 orx4 is added one obtains eitherx3 as the only equilibrium outcome.23 Then it cannot be an equilibrium to stop, as voters 1 or 3 would gain by proposing either x4 orx5.
The agendas that remain to be checked that might lead tox1 are those in
A(fx0; x1; x2g). Note that for any a02A(fx0; x1; x2; x3g), the outcome is x1, while for anya 2A(fx0; x1; x2; x5g) the outcomex4 orx5. Thus, consistency (CE3) implies that ifx1is an equilibrium outcome followingfx0; x1; x2g, then also x4 or x5 is an equilibrium outcome following fx0; x1; x2g, and that x1 can only come from proposing x3 next. Also, note thatx3is not an outcome under any agenda in A(fx0; x1; x2g) as it loses to x2, and also x2 and x0 are never outcomes under any agendas in A(fx0; x1; x2g). Then by (CE3) it follows that x1 is not an equilibrium outcome following fx0; x1; x2g, and those equilibrium outcomes are a subset of fx4; x5g.
The example uses the fact that agendas that lead to x1 must have x1 in one of the ¯rst three places in the agenda. This always leaves additional alternatives that can be proposed that would lead to other outcomes, and the preference for some of these other outcomes prevents the speci¯c agendas leading to x1 from being equilibrium agendas.
Thus, chaos is avoided and we have predictions that we end up inside a strict subset of the top cycle.
In fact, we also have a \lower bound" on the set of possible outcomes
22Explicitly, the agendas leading to an outcome of x 1
are fx0; x1; x2; x3; x5; x4g, fx0; x1; x2; x3; x4; x5g, fx0; x1; x2; x3; x4g, fx0; x1; x2; x4; x3g,
fx0; x1; x2; x3g,fx0; x1; x3; x2g,fx0; x1; x2g,fx0; x2; x1g, and fx0; x1g. 23By reasoning similar to that above, it is easily checked that ifx
5is added next, then
of sincere voting under sequential elimination - Claim 1 in the appendix implies that all Pareto optimal elements in the top cycle can be supported as outcomes of continuation equilibria.
Finally, we show that equilibrium agendas under sincere voting under sequential elimination can lead to Pareto ine±cient outcomes.
Example 5
Let X = fx0; x1; x2; x3; x4; x5g. The status quo is x0. There are 3 indi-viduals, with preferences given below.
² x1P1x2P1x5P1x3P1x4P1x0
² x5P2x3P2x4P2x1P2x2P1x0
² x4P3x1P3x5P3x2P3x3P1x0
The induced tournament T(P) is :
² x1 beatsx0, x2, x3, and x5,
² x2 beatsx0 and x3,
² x3 beatsx0 and x4,
² x4 beatsx0, x1 and x2,
² x5 beatsx0, x2, x3 and x4.
Note thatx2 is Pareto dominated by x1.
Next, note that a0 = (x0; x2; x3; x5; x4; x1) results in x5, which is voter 2's favorite. Thus, we know that it is possible to reach (x0; x2; x3). Then under (CE3), voter 1 is willing to propose x4 expecting the continuation of
a leading to x2, given that there is another continuation equilibrium leading tox5. As argued above, we then havea as an equilibrium continuation once (x0; x2; x3; x4) has been reached.
Thusx2 is an equilibrium outcome when the status quo is x0.
5.7
Handling In¯nities
Our discussion so far has focused on a ¯nite set of alternativesX. We now demonstrate how our analysis works in more general settings where the set of alternatives may be in¯nite. An important ¯rst remark is that the de¯nitions we have for continuation equilibria, (CE1)-(CE3), can be applied directly to the in¯nite case without modi¯cation.
However, there are new challenges that arise in applying the de¯nition of equilibria in in¯nite settings, which we will address below. One challenge is whether or not to de¯ne voting rules on in¯nite sequences of alternatives, and if it is done, how to do it. There are di®erent ways that this might be done and the speci¯c choice of how to do it is usually speci¯c to the setting in question. Another challenge is to establish existence of equilibrium sets. In the ¯nite case existence was straightforward as we could follow a simple backward induction argument. In the in¯nite case the issue is more subtle and will require using some characteristics of the setting being analyzed. A third challenge is that even when collections of sets of agenda equilibria can be shown to exist, it may still be hard to get a handle on a characterization of them as, again, a simply backward induction approach is precluded.
Nevertheless, despite these challenges the de¯nitions turn out to be quite manageable in several ways as we now show.
Consider an in¯niteX. LetA =[kAk be now the set of arbitrary length ¯nite agendas.24
24Here we could extend a voting rule V to be de¯ned over in¯nite agendas, but it is
not necessary. For the interested reader, one way of de¯ningV over in¯nite agendas is as follows. Consider an in¯nitea, and let akbe the agenda consisting of the ¯rstkproposed
alternatives. If there exists someK such that V(ak; P) =V(aK; P) for all k ¸ K, then
de¯neV(a; P) =V(aK; P). Have some rule for assigningV(a; P) otherwise, such as ¯xing
Given a voting rule V, say that an agenda a 2 A is maximal at P if
V(a0; P) =V(a; P) for alla02A(a). Denote the set of maximal agendas for
V and P that are the continuation of somea by MV(a; P). The analogue of Theorem 2 now follows.
First, we show that when the set of maximal agendas is nonempty, then there exists a natural set of continuation equilibria.
Lemma 1 Consider an in¯nite X, a pro¯le of preferences P 2 P¤, and a Pareto e±cient voting rule V such that MV(a; P) is nonempty for all agen-das a 2 A. Then there exists a collection of sets of continuation equilib-ria fCEV(a; P)ga2A satisfying (CE1)-(CE3), which is to set CEV(a; P) =
MV(a; P)for each a.
Lemma 1 leaves open the question of when MV(a; P) is nonempty for all agendas. This is easy to check in some cases as when there is a Condorcet winner, and can also be veri¯ed in some settings such as the three person divide-the-dollar game analyzed by Penn (2001). We leave the exploration of more subtle conditions guaranteeing nonemptyness for future research.
Now we can establish the analog of Theorem 2 for the in¯nite case.
Theorem 3 Consider an in¯nite X, a pro¯le of preferences P 2 P¤, and a Pareto e±cient voting ruleV such thatMV(a; P)is nonempty for all agendas
a2 A. For any collection of sets of continuation equilibria fCEV(a; P)ga2A (satisfying (CE1) and (CE2)), and any ¯nite a2A,
[a02CEV(a;P)V(a0; P)½ [b2MV(a;P)V(b; P);
and if consistency (CE3) is also satis¯ed, then
[a02CEV(a;P)V(a0; P) =[b2MV(a;P)V(b; P):
The proof of Theorem3 is provided in the appendix. Here, we provide the basic intuition. The proof of Theorem 2 exploited the possibility of backward induction from agendasa2Am. Notice that ifais a maximal agenda, then all
6
Discussion of the De¯nition of Equilibrium
Proposals to Stop the Agenda or Seconds to Continue an Agenda
Some procedures may allow an individual to propose a motion that voting take place immediately on the existing agenda. This motion is voted \yes" or \no", and a majority support can stop the existing agenda. Alternatively, a procedure may require at least two agents to support a proposal in order to add it to the agenda.
If either of these variations are present, it makes no di®erence to the analysis, at least under sophisticated voting by successive elimination. Let us o®er a heuristic argument for why Corollary 6 extends in this way.
We argue by induction. It is clearly true starting at some full length agenda. Suppose it is true starting at agendas of length at least k + 1. Consider an existing agenda a2 Ak, S(a; P) =x, and individual i proposes the motion that voting take place immediately. If i's motion is defeated, then her proposal is irrelevant. On the other hand, ifi's motion is accepted, thenx becomes the ¯naloutcome. This implies that a majority prefers x to any outcome that can be obtained by some further continuation equilibrium, which from the induction step and the corollary corresponds to the outcome of some a02 A(a)\Am. If x already corresponds to such an outcome, then the claim is true. If not, then by the Shepsle-Weingast algorithm, there must be some alternative y =2 a such that y is preferred by a majority to x and such thaty is the outcome under a continuation equilibriuma02A(a)\Am. This, however, implies that a majority would vote to continue rather than stop atx, which would be a contradiction. Thus the claim is true.
The argument for having a second agent move a proposal to make it part of an agenda is analogous, noting that if a majority prefer y to x, then at least two agents must prefer to follow the continuation equilibrium leading toy rather than stopping at x.
Modi¯cations of Consistency
We say that an agendaa0= (a; x; : : :) 2 CV+(a; P) isstrongly
rationaliz-able if there exists i2N such that for anyy =2a and y6=xthere exists some
a00 2 CEV((a; y); P) such that V(a0; P)RiV(a00; P), and if a 2 C EV(a; P) then alsoV(a0; P)RiV(a; P).
Strong rationalizability only allows for an agenda (a; x; : : :) which is a continuation ofa to be supported only if there is some agent who does not prefer all equilibrium continuations of (a; y) to those of (a; x). The idea being that an agent who prefers all continuations of (a; y) to those of (a; x) would not proposex, but would instead proposey (or possibly some other alterna-tive). This di®ers from rationalizability, in that rationalizability allows some
i to propose xif there is some alternative continuation that the agent ¯nds worse; but this does not consider the fact that the agent might prefer to propose something else iny's place.
We say that a collection of sets of continuation equilibria is strongly consistentif it satis¯es25
(CE4) (Strong Consistency) If a0 2 CV+(a; P) is strongly rationalizable, then
a02 CEV(a; P). Conversely, ifa0 = (a; x; : : :)2C EV(a; P) and either
a 2C EV(a; P) or a00 = (a; y; : : :)2 CEV(a; P) for somey 6=x, then a0 is strongly rationalizable.
Note that from Theorem 1 we know that for Pareto e±cient rules contin-uation equilibria satisfying strong consistency (CE4) always are a subset of those satisfying consistency (CE3). Example 1 is easily seen to be one where this is a strict subset. However, that is an ine±cient voting rule. The fol-lowing example shows that the selection may be strict even for sophisticated voting by successive elimination, where strong consistency results in a strict subset of the Banks' set.
Example 6
LetX=fx1; x2; x3; x4; x5g andN =f1;2;3g. The preference pro¯le is:
25When we modify (CE3) to (CE4), we might also consider adding another condition,
which was implied under (CE1), (CE2) and (CE3), but not under (CE1), (CE2) and (CE4). The condition is (5) If (a; x; : : :)2CEV(a; P) thenCEV((a; x); P)½CEV(a; P).
² x1P1x3P1x2P1x4P1x5
² x5P2x3P2x4P2x1P2x2
² x2P3x4P3x5P3x1P3x3
Then, the induced tournamentT(P) is
² x4 beatsx1 and x5.
² x1 beatsx2 and x3.
² x2 beatsx4 and x5.
² x3 beatsx2 and x4.
² x5 beatsx1 and x3.
Then, BS(fx0g; P) = X. We want to show that if CE(fx0g; P) satisfy (CE1),(CE2) and strong consistency, thenEOV(a; P) =fx1; x2; x5g.
First, note that if a 2 CS+(fx0; x1g; P), then S(a; P) = x4. For if a 2
A(fx1g), the possible outcomes are in fx5; x4g. But sincex4beatsx5, (CE2) implies thatS(a; P) =x4 if a2CS+(fx0; x1g; P).
Analogously, the following are true.
² If a 2CS+(fx0; x2g; P), then S(a; P) =x1.
² If a 2CS+(fx0; x5g), then S(a; P) =x2.
² If a 2CS+(fx0; x3g), then S(a; P) =x5.
² If a 2CS+(fx0; x4g; P), then S(a; P) =x3.
The proof is completed by showing that no one wants to propose x1 or
x4 initially.
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Appendix
Let
EOV(a; P) =[a02CEV(a;P)V(a0; P):
We ¯rst state and prove a lemma which will be used repeatedly.
Lemma 2 Consider anyP 2 P anda 2A. Suppose that[b2A(a)\AmV(b; P) ½
D(P), for some D(P)½X such that if x; y 2X, and x =2D(P); y 2 D(P)
then9i2N such that yPix. Under (CE1) and (CE2)EOV(a; P)½D(P).
Proof of Lemma 2: We prove this by induction on the cardinality of a. If
a2Am, thenC E
V(a; P) =fag, and so the assertion must be true. Suppose that for some K < m, the claim is true for each k > K and a 2 Ak. We show that the claim is true fora 2AK.
From the induction hypothesis it follows that CEV+(a; P) ½ D(P), and so from (CE1) we only need to show that if V(a; P) 2= D(P), then a =2 C EV(a; P). Consider any x =2 a, and b 2 C EV((a; x); P) ½ D(P). Since
V(a; P)2= D(P), it follows from the properties ofD(P) thatV(b; P)PiV(a; P) for somei. (CE2) then implies that a =2CEV(a; P), as required.
Proof of Theorem 1: Fix a Pareto e±cient V and a pro¯le P. The proof that V(a0; P) is Pareto e±cient for any a0 in CE
V(a; P) and
a2 A follows directly from Lemma 2, by letting D(P) in the lemma be the set of Pareto e±cient alternatives inX.
To complete the proof of the theorem, we show that (CE1), (CE2) and (CE3) can be satis¯ed if and only if26
C EV(a; P) = (
P EV(a; P) if V(a; P)RiV(a0; P) for alli and a0 2CV+(a; P)
P EV(a; P)na if V(a0; P)PiV(a; P) for some iand a02CV+(a; P). It is straightforward to check if CEV(a; P) is de¯ned above then (CE1),
(CE2) and (CE3) are satis¯ed. So we show the converse.
ConsiderCEV(a; P) satisfying (CE1), (CE2) and (CE3). The proof pro-ceeds by induction. Note that for any a 2 Am, C E
V(a; P) = fag and that by the Pareto e±ciency of V the claim follows directly. So, consider some
26Note that in the second case it must be that P E
